To find a gap, write out one solved question and explain every line in your own words until you reach a line you cannot justify. That line, not the whole topic, is what needs repair. Most gaps are one small rule or one old skill, not a whole chapter.
This works for any subject with worked steps: maths, sciences, accounting. The example below uses algebra because the steps are easy to see.
Why does “I don’t get this topic” hide the real problem?
A topic is a chain of small moves. If one link is weak, the whole chain feels impossible, and you can only say “I don’t get it”. That sentence is too big for anyone to fix quickly.
A sentence like “I don’t know why the 3 multiplies the 2 as well as the x” is small enough to fix in a few minutes. Your job is to shrink the problem until it can be said in one sentence.
How do I explain each step and find the gap?
- Pick one fully worked solution from your notes or textbook, ideally one you got wrong earlier.
- Cover everything below the first line. Read only the question and the first line of working.
- Say aloud why that line follows. Use words like “because” and “so”. If you can only say “that is what the teacher did”, you have found a weak step.
- Reveal the next line and repeat. Stop at the first line where you hesitate.
- Write the gap as a question starting with “why” or “how do I know”.
Worked example
Solve 3(x + 2) = 2x + 9.
The worked solution in the notes reads:
- Line 1: 3x + 6 = 2x + 9
- Line 2: x + 6 = 9
- Line 3: x = 3
Say each line aloud. Line 2 is fine: subtract 2x from both sides, so 3x − 2x = x. Line 3 is fine: subtract 6 from both sides. But line 1 makes a student stop: why did the 2 become 6?
The gap is not “algebra”. It is “the 3 outside the bracket multiplies both terms inside”. Once that is clear, the rest works.
Check the answer: 3(3 + 2) = 15 and 2(3) + 9 = 15, so x = 3 is correct.
What is a common mistake when doing this?
The most common mistake is to read the working silently and decide “yes, that looks right”. Recognising a line is not the same as explaining it. You can recognise line 1 and still write 3x + 2 = 2x + 9 in your own attempt.
Mistaken attempt: 3(x + 2) = 2x + 9 becomes 3x + 2 = 2x + 9, so x = 7.
The 3 was applied to x only. The check shows the problem: 3(7 + 2) = 27, but 2(7) + 9 = 23.
The correction is to put the check in the habit. Substituting your answer back takes ten seconds and shows whether a hidden gap is still there.
Check yourself
In each solution, find the first line that is wrong and say why.
1. Solve 5 − 2(x − 1) = 11.
- Line 1: 3(x − 1) = 11
- Line 2: 3x − 3 = 11
- Line 3: x = 14/3
Show answer
Line 1 is the first wrong step. The 5 and the 2 cannot be combined, because 2 multiplies the bracket and is not subtracted from 5 first. Correct working: 5 − 2x + 2 = 11, so 7 − 2x = 11, so −2x = 4 and x = −2. Check: 5 − 2(−3) = 11.
2. Work out 1/2 + 1/3.
- Line 1: = 2/5
Show answer
Line 1 is wrong: the numerators and denominators were added separately. Use a common denominator: 3/6 + 2/6 = 5/6.
3. A price of RM50 rises by 20%, then the new price falls by 20%. The working says 50 + 10 = 60, then 60 − 10 = 50. Which step is wrong?
Show answer
The second step. 20% of 60 is 12, not 10, because the percentage is now taken of the new amount. The correct final price is 60 − 12 = RM48.
What do I do once I have found the gap?
Write the gap as one question and look for a short, focused explanation of that single idea. Then try two new questions that use it, with different numbers. The prerequisite gap explorer can suggest which earlier skill to look at if a wrong answer surprises you.
If you want a structure for the next step, read preparing basic questions for a one-to-one lesson and building a repair plan alongside current classwork. You can also see the wider students’ guide.
Some gaps close with an hour of careful self-study. When the same kind of gap keeps returning, or you cannot tell whether your explanation is right, a teacher can listen to your reasoning and spot the slip. That is what online one-to-one Mathematics tuition is designed for.